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2 Journal of Applied Analysis

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Universally polygonally approximable functions

100%

Evans. M. J. , Humke P. D. , O`Malley R. J.

2000

tom Vol. 6, nr 1

25-45

It has recently been established that any Baire class one function f : [0,1] -> R can be represented as the pointwise limit of a sequence of polygonal functions whose vertices lie on the graph of f. Here we investigate the subclass of Baire class one functions having the additional property that for every dense subset D of [0,1], the first coordinates of the vertices of the polygonal functions can be chosen from D.

Exceptional sets for universally polygonally approximable functions

84%

Evans M. J. , Humke P. D.

tom Vol. 7, nr 2

175-190

In [9], the present authors and Richard O'Malley showed that in order for a function be universally polygonally approximate it is necessary that for each ε > 0, the set of points of non-quasicontinuity be σ - (1 - ε ) symmetrically porous. The question as to whether that condition is sufficient or not was left open. Here we prove that if a set, E = U∞n=1 En, such that each Ei is closed and 1-symmetrically porous, then there is a universally polygonally approximable function, f, whose set of points of non-quasicontinuity is precisely E. Although it is tempting to call this a partial converse to our earlier theorem it might be more since it is not known if these two notions of symmetric porosity differ in the class of F? sets.